Various fuel injectors are known, including solenoid actuated fuel injectors and piezoelectrically actuated fuel injectors. Compared with solenoid actuated fuel injectors, piezoelectrically actuated injectors have a higher bandwidth, which allows for the delivery of more complex injection rate profiles, examples including tightly-spaced pulse trains and rate shaping. As is known in the art, injection rate shaping may reduce overall fuel consumption and improve the trade-off between NOx and particulate matter emissions.
A boot shape injection profile is depicted in FIG. 1, and is an example of rate shaping. Profile 100 includes a “toe” 102 and a “shank” 104. Profile 100 may provide benefits for diesel engines operating at high load and medium speed. Various techniques may be employed for rate shaping. In U.S. Pat. No. 6,079,641, a piezoelectric fuel injector with open-loop control is disclosed for producing rate shaped injections. In Kohketsu, S., Tanabe, K., and Mori, K., 2000, “Flexibly controlled injection rate shape with next generation common rail system for heavy duty DI diesel engines,” SAE Technical Paper (2000-01-0705), a system with two common rails is disclosed for creating rate shaped injection profiles. In U.S. Pat. No. 7,896,257, a position sensor is disclosed for estimating fueling rate for the purpose of closed-loop injection rate control and failure diagnosis. In Wu, C., and Sun, Z., 2013, “Design and control of a direct fuel injector with rate shaping capability,” American Control Conference 2013, Washington, D.C., an injector design is outlined which can enable rate shaping by utilizing an internal feedback mechanism.